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SUMMARY:On the arithmetic of Calabi-Yau manifolds: periods\, zeta function
s and attractor varieties
DTSTART;VALUE=DATE-TIME:20210604T100000Z
DTEND;VALUE=DATE-TIME:20210604T111500Z
DTSTAMP;VALUE=DATE-TIME:20240224T160415Z
UID:indico-contribution-5739@indico.ipmu.jp
DESCRIPTION:Speakers: Xenia de la Ossa (Oxford U.)\nIn this seminar I will
discuss the arithmetic of Calabi-Yau 3-folds. The main goal is to explore
whether there are questions of common interest in this context to physici
sts\, number theorists and geometers. The main quantities of interest in
the arithmetic context are the numbers of points of the manifold considere
d as a variety over a finite field. We are interested in the computation o
f these numbers and their dependence on the moduli of the variety. The su
rprise for a physicist is that the numbers of points over a finite field a
re also given by expression that involve the periods of a manifold. The nu
mber of points are encoded in the local zeta function\, about which much i
s known in virtue of the Weil conjectures. I will discuss interesting t
opics related to the zeta function and the appearance of modularity for on
e parameter families of Calabi-Yau manifolds.\nA topic I will stress is th
at for these families there are values of the parameter for which the mani
fold becomes singular and for these values the zeta function degenerates a
nd exhibits modular behaviour. I will report (on joint work with Philip
Candelas\, Mohamed Elmi and Duco van Straten) on an example for which the
quartic numerator of the zeta function factorises into two quadrics at sp
ecial values of the parameter which satisfy an algebraic equation with coe
fficients in Q (so independent of any particular prime)\, and for which th
e underlying manifold is smooth. We note that these factorisations are du
e to a splitting of the Hodge structure and that these special values of t
he parameter are rank two attractor points in the sense of type IIB superg
ravity. Modular groups and modular forms arise in relation to these attra
ctor points.\nTo our knowledge\, the rank two attractor points that were f
ound by the application of these number theoretic techniques\, provide the
first explicit examples of such points for Calabi-Yau manifolds of full S
U(3) holonomy.\n\nhttps://indico.ipmu.jp/event/387/contributions/5739/
LOCATION:Online
URL:https://indico.ipmu.jp/event/387/contributions/5739/
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